Here is the full list of all 22 Dog Duty achievements.
Finished the game
Destroyed the Octo Plane
Rescued Paula
Destroyed the Octo Tank
Destroyed the Octo Jumper
Rescued Rebecca
Stacked a good amount of enemy goods
Rescued Madness
Found a deeper layer
Cleared all locations
Conquered all armor depots.
Destroyed all enemy communications devices.
Won a battle with every vehicle.
Defeated an enemy squadron on the Hoovercraft.
The game finished before even starting! How innovative!
Defeated an enemy squadron on the Boat.
Defeated an enemy squadron on the Truck.
Found an abandoned car in the Desert.
Your vehicle has been armed.
Rescued Kilmer's bootleg character.
Rescued Liza.